Quantitative Methods

1. Definition of Interest Rate

The interest rate $r$ can be interpreted as:

1. A required rate of return: The investor’s required compensation.
2. A discount rate: The rate used to determine the present value of future cash flows.
3. An opportunity cost: The return foregone by investing in a particular asset instead of the next best alternative.

These reflect the relationship between cashflows.

2. The Interest Rate and Time Value of Money

$$\text{Interest rate} = \text{Risk-free rate} + \text{Premiums (bearing distinct types of risk)}$$

Also called the yield.

Example: $9500 today equals $10000 in one year → this is the discount rate.

$$\frac{10{,}000 - 9{,}500}{9{,}500} = \frac{500}{9{,}500} \approx 5.26% \quad \text{(rate of return)}$$

2.1 Determinants of Interest Rates

$$r = r_{RF} + \text{Inflation premium} + \text{Default risk premium} + \text{Liquidity premium} + \text{Maturity premium}$$

2.2 Nominal vs. Real Risk-Free Rate

The nominal risk-free interest rate reflects both the real risk-free rate and the inflation premium.

Fisher equation (exact):

$$1 + i = (1 + r_{RF})(1 + \pi)$$

$r_{RF}$: Nominal risk-free rate

$\pi$: Inflation rate

Approximation (when $r_{RF}$ and $\pi$ are small):

$$i \approx r_{RF} + \pi$$

since $(1 + r_{RF})(1 + \pi) = 1 + r_{RF} + \pi + r_{RF}\pi \approx 1 + r_{RF} + \pi$.

as $r_{RF}\pi$ is very small and can be ignored.

3. Rates of Return

Calculate and interpret different approaches to return measurement over time and describe their appropriate uses.

Financial assets generate returns through:

graph TD
    A[Financial Asset Return] --> B[Periodic Income]
    A --> C[Price Movement]
    B --> D[Cash dividends]
    B --> E[Interest payments]
    B --> F[Pension plans / retirement annuities]
    C --> G[Capital gain / loss]
    C --> H[Non-dividend-paying stocks]

3.1 Holding Period Return

$$R = \frac{P_1 - P_0 + I_1}{P_0}$$

where $P$ is the price and $I_1$ is the income; we buy at time 0 and sell at time 1.

For periods longer than 1 year, with annual returns $R_1, R_2, R_3$:

$$R = \left[(1+R_1)(1+R_2)(1+R_3)\right] - 1$$

3.2 Arithmetic Mean Return

For holding-period returns (daily, monthly, annual), the arithmetic mean (mean return) is:

$$\bar{R}i = \frac{R{i,1} + R_{i,2} + \cdots + R_{i,T}}{T} = \frac{1}{T}\sum_{t=1}^{T} R_{i,t}$$

Assumes the amount invested at the beginning of each period is the same.

3.3 Geometric Mean Return

$$\bar{R}{G,i} = \sqrt[T]{\prod{t=1}^{T}(1 + R_{i,t})} - 1$$

Example:

$$\sqrt[3]{(1-0.50)(1+0.35)(1+0.27)} - 1 \approx -0.0500 \quad \text{(more precise)}$$

The geometric mean is always less than or equal to the arithmetic mean.
The two are equal only when there is no variability in the observations.

3.4 Harmonic Mean

$$\bar{X}H = \frac{n}{\displaystyle\sum{i=1}^{n}\frac{1}{X_i}}, \quad X_i > 0 \text{ for } i = 1, 2, \ldots, n$$

Example: for ${1, 2, 3, 4, 5, 6, 1000}$, $\bar{X}_H = 2.8560$ (outlier has less impact).

Properties:

• Measure of central tendency in the presence of outliers
• Appropriate for averaging ratios (“amount per unit”)
• Works only for non-negative numbers

Key inequality:

$$\bar{X}_A \geq \bar{X}_G \geq \bar{X}_H$$

$$\bar{X}_A \times \bar{X}_H = \bar{X}_G^2$$

Mean speed example (travel from A to B at speed $v_1$, return at speed $v_2$):

$$\text{mean speed} = \frac{2}{\dfrac{1}{v_1} + \dfrac{1}{v_2}} = \frac{2v_1 v_2}{v_1 + v_2}$$

3.5 Trimmed and Winsorized Means

Both reduce outlier impact while retaining all data points (Winsorized) or sample size (Trimmed).